Abstract: For the flag variety of a reductive algebraic group we define and describe explicitly a certain (set-theoretical) cross-section . The definition of depends only on a choice of reduced expression for the longest element in the Weyl group . It assigns to any a representative together with a factorization into simple root subgroups and simple reflections. The cross-section is continuous along the components of Deodhar's decomposition of . We introduce a generalization of the Chamber Ansatz and give formulas for the factors of . These results are then applied to parametrize explicitly the components of the totally nonnegative part of the flag variety defined by Lusztig, giving a new proof of Lusztig's conjectured cell decomposition of . We also give minimal sets of inequalities describing these cells.

R. J. MarshAffiliation:
Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH
Address at time of publication:
Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH
Email:
rjm25@mcs.le.ac.uk

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